Methods for producing holes with side-delimiting flanks are known in the prior art. So, for example, a method for laser-assisted formation of cooling air holes in turbine blades is described in U.S. Pat. No. 6,420,677. In this case, provision is made to discharge a sequence of laser pulses onto the turbine blade surface, wherein parts of the turbine material are vaporized so that a hole is formed along a Z-axis. The laser beam is discharged onto the blade surface in an inclined manner by angles of +/−10-20° in relation to the Z-axis during the machining.